Stripe to spot transition in a plant root hair initiation model

A generalised Schnakenberg reaction-diffusion system with source and loss terms and a spatially dependent coefficient of the nonlinear term is studied both numerically and analytically in two spatial dimensions. The system has been proposed as a model of hair initiation in the epidermal cells of plant roots. Specifically the model captures the kinetics of a small G-protein ROP, which can occur in active and inactive forms, and whose activation is believed to be mediated by a gradient of the plant hormone auxin. Here the model is made more realistic with the inclusion of a transverse co-ordinate. Localised stripe-like solutions of active ROP occur for high enough total auxin concentration and lie on a complex bifurcation diagram of single and multi-pulse solutions. Transverse stability computations, confirmed by numerical simulation show that, apart from a boundary stripe, these 1D solutions typically undergo a transverse instability into spots. The spots so formed typically drift and undergo secondary instabilities such as spot replication. A novel 2D numerical continuation analysis is performed that shows the various stable hybrid spot-like states can coexist. The parameter values studied lead to a natural singularly perturbed, so-called semi-strong interaction regime. This scaling enables an analytical explanation of the initial instability, by describing the dispersion relation of a certain non-local eigenvalue problem. The analytical results are found to agree favourably with the numerics. Possible biological implications of the results are discussed.Comment: 28 pages, 44 figure

Laguerre polynomials of derivations

We introduce a grading switching for arbitrary non-associative algebras of prime characteristic p, aimed at producing a new grading of an algebra from a given one. We take inspiration from a fundamental tool in the classification theory of modular Lie algebras known as toral switching, which relies on a delicate adaptation of the exponential of a derivation. Our grading switching is achieved by evaluating certain generalized Laguerre polynomials of degree p − 1, which play the role of generalized exponentials, on a derivation of the algebra. A crucial part of our argument is establishing a congruence for them which is an appropriate analogue of the functional equation exp(x) · exp(y) = exp(x+y) for the classical exponential. Besides having a wider scope, our treatment provides a more transparent explanation of some aspects of the original toral switching, which can be recovered as a special case

Pseudo-plateau bursting and mixed-mode oscillations in a model of developing inner hair cells

This is the final version. Available on open access from Elsevier via the DOI in this recordInner hair cells (IHCs) are excitable sensory cells in the inner ear that encode acoustic information. Before the onset of hearing IHCs fire calcium-based action potentials that trigger transmitter release onto developing spiral ganglion neurones. There is accumulating experimental evidence that these spontaneous firing patterns are associated with maturation of the IHC synapses and hence involved in the development of hearing. The dynamics organising the IHCs’ electrical activity are therefore of interest. Building on our previous modelling work we propose a three-dimensional, reduced IHC model and carry out non-dimensionalisation. We show that there is a significant range of parameter values for which the dynamics of the reduced (three-dimensional) model map well onto the dynamics observed in the original biophysical (four-dimensional) IHC model. By estimating the typical time scales of the variables in the reduced IHC model we demonstrate that this model could be characterised by two fast and one slow or one fast and two slow variables depending on biophysically relevant parameters that control the dynamics. Specifically, we investigate how changes in the conductance of the voltage-gated calcium channels as well as the parameter corresponding to the fraction of free cytosolic calcium concentration in the model affect the oscillatory model bahaviour leading to transition from pseudo-plateau bursting to mixed-mode oscillations. Hence, using fast-slow analysis we are able to further our understanding of this model and reveal a path in the parameter space connecting pseudo-plateau bursting and mixed-mode oscillations by varying a single parameter in the model.Engineering and Physical Sciences Research Council (EPSRC

Growth induction and low-oxygen apoptosis inhibition of human CD34 + progenitors in collagen gels

Various reports have indicated low survival of injected progenitors into unfavorable environments such as the ischemic myocardium or lower limb tissues. This represents a major bottleneck in stem-cell-based cardiovascular regenerative medicine. Strategies to enhance survival of these cells in recipient tissues have been therefore sought to improve stem cell survival and ensure long-term engraftment. In the present contribution, we show that embedding human cord blood-derived CD34+ cells into a collagen I-based hydrogel containing cytokines is a suitable strategy to promote stem cell proliferation and protect these cells from anoxia-induced apoptosis

Secondary Vitrectomy with Internal Limiting Membrane Plug due to Persistent Full-Thickness Macular Hole OCT-Angiography and Microperimetry Features: Case Series

Purpose. To study the features in OCT-angiography and microperimetry in eyes with persistent full-thickness macular hole (FTMH) closed with the secondary plana vitrectomy (PPV) with autologous internal limiting membrane (ILM) plug. Methods. Secondary PPV was performed with closing the persistent FTMH with ILM plug, C3F8 tamponade, and face-down positioning. Four patients were followed for 6 months with best corrected visual acuity (BCVA) measurement, SD-OCT and OCT-A, and microperimetry. The results were compared with the fellow eye; in two patients, it was the healthy eye, and in two remaining eyes, successfully closed FTMH after primary PPV. Results. ILM flap was integrated in all cases with V-shape of closure, and atrophy was found in one case, with the largest diameter of FTMH. BCVA improved in two cases and remained the same in two cases. In OCT-A, the area of foveal avascular zone (FAZ) was larger, and foveal vessel density (FVDS) was smaller in eyes after secondary PPV in comparison to fellow eyes. In microperimetry, retinal sensitivity was lower in eyes after secondary PPV, and eccentric fixation was found in 2 of 4 patients. Conclusion. Although the anatomical results of repeated surgeries of FTMH with ILM plug are favorable, visual function results may be limited. Secondary closure of FTMH with ILM plug may lead to atrophy, changes in the macular vasculature, and eccentric fixation. The trial is registered with NCT03701542

A generalized truncated logarithm

We introduce a generalization $G^{(\alpha)}(X)$ of the truncated logarithm $\pounds_1(X)=\sum_{k=1}^{p-1}X^k/k$ in prime characteristic $p$, which depends on a parameter $\alpha$. The main motivation of this study is $G^{(\alpha)}(X)$ being an inverse, in an appropriate sense, of a parametrized generalization of the truncated exponential given by certain Laguerre polynomials. Such Laguerre polynomials play a role in a {\em grading switching} technique for non-associative algebras, previously developed by the authors, because they satisfy a weak analogue of the functional equation $\exp(X)\exp(Y)=\exp(X+Y)$ of the exponential series. We also investigate functional equations satisfied by $G^{(\alpha)}(X)$ motivated by known functional equations for $\pounds_1(X)=-G^{(0)}(X)$